Final answer:
A common time base in a coupled circuit dictates the equality of the voltage amplitudes (VA) in all parts of the circuit due to conservation of charge, and the base mutual inductance must be the geometric mean of the self-inductance bases to align with the common time base.
Step-by-step explanation:
To show that the choice of a common time base in a coupled circuit automatically enforces the equality of voltage amplitude (VA) across all parts of the circuit, it's important to first understand that conservation of charge requires the currents in different components of an LC (inductor-capacitor) circuit, such as resistors (R), inductors (L), and capacitors (C), to be the same and in phase. This implies that the voltage amplitudes must adjust to satisfy Kirchhoff's loop rule, which states that the total voltage in a loop must equal the source voltage at all times. Consequently, when components are coupled through mutual inductance, the induced electromotive forces (emfs) need to align with this common time base.
The mutual inductance (M) is a measure of how effectively two circuits are coupled, which is largely a geometric characteristic independent of the coil currents. If inductors with self-inductances L1 and L2 are chosen such that they share a common geometric mean with regards to the base mutual inductance M12B, (which is set to be the geometric mean of the self-inductance bases, i.e., M12B = (L1B × L2B)⁰.⁵), then the self-inductances must be in perfect symmetrical relation to one another in order to maintain the common time base and voltage amplitudes.